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Citation: Enfedaque, A.; Suárez, F.;

Alberti, M.G.; Gálvez, J.C. Suitability

of Constitutive Models of the

Structural Concrete Codes When

Applied to Polyolefin Fibre

Reinforced Concrete. Materials 2022,

15, 2323. https://doi.org/

10.3390/ma15062323

Academic Editor: Dario De

Domenico

Received: 8 February 2022

Accepted: 16 March 2022

Published: 21 March 2022

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materials

Article

Suitability of Constitutive Models of the Structural ConcreteCodes When Applied to Polyolefin Fibre Reinforced ConcreteAlejandro Enfedaque 1 , Fernando Suárez 2 , Marcos G. Alberti 1 and Jaime C. Gálvez 1,*

1 Departamento de Ingeniería Civil-Construcción, Universidad Politécnica de Madrid, E.T.S.I. Caminos,Canales y Puertos, 28040 Madrid, Spain; alejandro.enfedaque@upm.es (A.E.); marcos.garcia@upm.es (M.G.A.)

2 Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, 23071 Jaén, Spain; fsuarez@ujaen.es* Correspondence: jaime.galvez@upm.es; Tel.: +34-910-674-125

Abstract: The use of fibres as structural reinforcement in concrete is included in standards, providingguidelines to reproduce their behaviour, which have been proven adequate when steel fibres are used.Nevertheless, in recent years new materials, such as polyolefin fibres, have undergone significantdevelopment as concrete reinforcement. This work gives insight on how suitable the constitutivemodels proposed by the Model Code 2010 (MC2010) are in the case of such polymer fibres. A setof numerical models has been carried out to reproduce the material behaviour proposed by theMC2010 and the approach based on the softening function proposed by the authors. The results showremarkable differences between the experimental results and the numerical simulations when theconstitutive models described in the MC2010 are employed for different polyolefin fibre reinforcedconcrete mixes, while the material behaviour can be reproduced with greater accuracy if the softeningfunction proposed by the authors is employed when this type of macro-polymer fibres is used.Moreover, the relatively complex behaviour of polyolefin fibre reinforced concrete may be reproducedby using such constitutive model.

Keywords: fibre-reinforced concrete; polymeric fibres; Model Code 2010; cohesive model

1. Introduction

Concrete has been one of the most employed construction material [1] in modernsocieties in spite of its impact on the environment [2]. Such use has been based on itsresistance to natural weather conditions [3,4] and its remarkable compressive strengthand modulus of elasticity [5,6]. However, plain concrete is also known as a quasi-brittlematerial which is not capable of sustaining high-tensile stresses. Adding reinforcing barsof concrete to form reinforced concrete was one of the possible solutions to this problem.Another possibility is adding fibres randomly distributed while mixing, forming fibrereinforced concrete (FRC). The effect of the addition of fibres in the mechanical properties ofconcrete depends on numerous parameters, such as the type of fibres used, dosage, shapeand size, distribution, orientation and, among others, pull-out response [7,8]. Addingcertain types of fibres in a determined volume fraction to concrete a material suitablefor numerous applications can be obtained. For instance, steel fibre reinforced concrete(SFRC) has been successfully applied to pavements [9,10], tunnel linings [11,12], or evenseismic structures [13,14]. However, the use of SFRC might have two main drawbacks,its durability under certain circumstances [15,16] and its environmental impact [17]. Onepossibility to overcome such durability issues is to use new polymer macro-structuralfibres, which are chemically stable when applied in foundations, tunnel linings or marineenvironments [18–21]. Polyolefin fibre reinforced concrete (PFRC) is one of the materialsavailable for these uses.

All the previous applications have been possible due to several national codes andrecommendations [22–27]. These documents have set the conditions that a certain FRC

Materials 2022, 15, 2323. https://doi.org/10.3390/ma15062323 https://www.mdpi.com/journal/materials

Materials 2022, 15, 2323 2 of 15

should meet so that the improvements provided by the fibres added might enable areduction in the amount of steel reinforcing bars used. Such requirements take into accountthe residual strength of the FRC studied at certain crack openings which are related to theservice limit state (SLS) and the ultimate limit state (ULS). For checking if a FRC formulationboasts sufficient residual strength, bending tests are performed on FRC specimens inlaboratory conditions. Once such tests have been performed, and if the FRC propertiesexceed the requirements set, the contribution of the fibres might be included in the structuraldesign of the concrete element. Although it is true that both SFRC and PFRC have beenshown as suitable for their application in structural concrete elements [7,28,29] it should notbe overlooked that the extensive experience related with SFRC served as reference whenthe requirements of the material were set. Once such requirements have been checked,the aforementioned recommendations propose several constitutive models that should beapplied to the structural design and that have been successfully applied when using SFRC.Similarly, the constitutive models proposed used several characteristics that were inspiredby the mechanical behaviour of SFRC. Consequently, the accuracy of such approacheswhen dealing with PFRC is a matter that deserves being studied.

The behaviour of PFRC can meet the requirements in the standards [30] although theshapes of the fracture curves have different features compared with those of SFRC [29].According to the Model Code 2010 (MC2010) [31], the first condition that FRC shouldmeet is a minimum load value, proportional to the maximum strength in the limit ofproportionality, at a crack mouth opening displacement of 0.5 mm; this limitation isintended to prevent brittle behaviour of the material. Although PFRC with an average fibredosage might meet the cited requirement, polyolefin fibres need greater deformations toabsorb the load that concrete is not capable of sustaining being the first unloading branchsteeper than in SFRC. The second condition that MC2010 establishes is that at a crack mouthopening displacement of 2.5 mm the load must be over 20% of the strength at the limitof proportionality. This limitation for PFRC is met even for very low fibre dosages [30].As there are remarkable differences in the post peak mechanical behaviours of SFRC andPFRC, the applicability of the constitutive models, which are built based on certain valuesof their residual strength, should be studied. Thus, this study seeks to address the mainissues concerning the considerations in the standards regarding the mechanical behaviourof PFRC.

The main goals of this contribution are the following. First, an implementation of thelinear and rigid-plastic models developed in MC2010 in a numerical code by means ofa material subroutine is sought. Second, the differences between the material behaviourobtained with the constitutive models of MC2010 and those obtained using the trilinearapproach proposed by the authors that delivers an accurate reproduction of the PFRCbehaviour are assessed. This matter will be carried out in four mixes of PFRC with 3, 4.5,6, and 10 kg/m3 of polyolefin fibres. Lastly, an evaluation of the material behaviour atSLS and ULS and another regarding the fracture energy consumed in the tests are carriedout for determining the reliability of using the studied approaches. Summarising, thismanuscript supplies additional information for a possible future adaptation of MC2010,and other national concrete structural codes based on it when applied to PFRC.

2. Post-Peak Material Modelling

In this section, the numerical modelling of fracture used in the simulations is addressed.In the first part, the embedded cohesive crack model is briefly presented. Since the readercan find a detailed description in previous works published by the authors, this descriptionhas been kept to a minimum. In the second part, the model is adapted for PFRC followingthe guidelines of the MC2010.

2.1. Embedded Cohesive Crack Modeling

Fracture is modelled by means of a cohesive crack model, based on the work proposedby Hillerborg [32] and implemented as an embedded cohesive crack developed in a tri-

Materials 2022, 15, 2323 3 of 15

angular finite element. This numerical modelling has been used for simulating fractureof concrete [33] and has been adapted for use with brick masonry [34] and, more recently,with PFRC [35].

In this model, fracture is assumed to develop mainly under mode I and with thecohesive stress vector t being parallel to the displacement vector w (thus, it is a centralforce model), as expressed by (1):

t =f(∣∣∣ ~

w∣∣∣)∣∣∣ ~

w∣∣∣ w with

~w = max(|w|) (1)

with f( ~

w)

being the softening function, expressed in terms of~w, which corresponds to an

equivalent crack opening that considers the maximum historical opening, necessary forcorrectly reproducing possible loading-unloading situations.

This formulation is implemented for triangular elements and, as cracks can onlydevelop in lines parallel to the sides of the element, only three cracking orientations areallowed (see Figure 1). It follows a strong discontinuity approach, dividing the elementinto two parts, A+ and A−, with t being parallel to the crack displacement and constantalong the length of the crack L. It is obtained by (2):

t =AhL

σ·n (2)

with A being the area of the element, h the height of the triangle over the side opposite tothe solitary node and n the unit vector normal to the crack (see Figure 1).

Materials 2022, 15, x FOR PEER REVIEW 3 of 15

2.1. Embedded Cohesive Crack Modeling Fracture is modelled by means of a cohesive crack model, based on the work pro-

posed by Hillerborg [32] and implemented as an embedded cohesive crack developed in a triangular finite element. This numerical modelling has been used for simulating frac-ture of concrete [33] and has been adapted for use with brick masonry [34] and, more recently, with PFRC [35].

In this model, fracture is assumed to develop mainly under mode I and with the co-hesive stress vector 𝐭 being parallel to the displacement vector 𝐰 (thus, it is a central force model), as expressed by (1): 𝐭 = 𝑓(|𝐰|)|𝐰| 𝐰 𝑤𝑖𝑡ℎ 𝐰 = 𝑚𝑎𝑥(|𝐰|) (1)

with 𝑓(𝐰) being the softening function, expressed in terms of 𝐰, which corresponds to an equivalent crack opening that considers the maximum historical opening, necessary for correctly reproducing possible loading-unloading situations.

This formulation is implemented for triangular elements and, as cracks can only de-velop in lines parallel to the sides of the element, only three cracking orientations are al-lowed (see Figure 1). It follows a strong discontinuity approach, dividing the element into two parts, A+ and A−, with 𝐭 being parallel to the crack displacement and constant along the length of the crack 𝐿. It is obtained by (2): 𝐭 = 𝐴ℎ𝐿 𝛔 · 𝐧 (2)

with 𝐴 being the area of the element, ℎ the height of the triangle over the side opposite to the solitary node and 𝐧 the unit vector normal to the crack (see Figure 1).

Figure 1. Possible crack paths (left) and geometrical definitions of 𝒘, 𝐧 and 𝐛 (right).

The material around the crack is elastic and the stress tensor is obtained by using a strain vector that results from subtracting the strain due to the crack displacement from the apparent strain 𝜀 , obtained under elastic considerations and by using the nodal dis-placements (Equation (3)): 𝛔 = 𝐄: 𝜀 − (𝐛 ⨂𝐰) (3)

where 𝐄 stands for the elastic tangent tensor, 𝜀 for the apparent strain vector obtained with the nodal displacements, 𝐛 for the gradient vector of the shape function of the sol-itary node, which can be obtained as: 𝐛 = 1ℎ 𝐧 (4)

Superscript S makes reference to the symmetric part of the tensor, the usual double-dot product ((𝐴: 𝐵) = 𝐴 𝑏 ) and ⨂ the usual direct product ((𝒂⨂ 𝒃)𝒊𝒋 = 𝒂𝒊𝒃𝒋).

Lastly, by using the expression of the stress tensor (3), the stress vector can be ob-tained as:

Figure 1. Possible crack paths (left) and geometrical definitions of w, n and b+ (right).

The material around the crack is elastic and the stress tensor is obtained by usinga strain vector that results from subtracting the strain due to the crack displacementfrom the apparent strain εa, obtained under elastic considerations and by using the nodaldisplacements (Equation (3)):

σ = E :[

εa −(

b+⊗

w)S]

(3)

where E stands for the elastic tangent tensor, εa for the apparent strain vector obtained withthe nodal displacements, b+ for the gradient vector of the shape function of the solitarynode, which can be obtained as:

b+ =1h

n (4)

Superscript S makes reference to the symmetric part of the tensor, the usual double-dotproduct ((A : B)ij = Aijklbkl) and

⊗the usual direct product ((a

⊗b)ij = aibj).

Materials 2022, 15, 2323 4 of 15

Lastly, by using the expression of the stress tensor (3), the stress vector can be obtained as:

t =f(∣∣∣ ~

w∣∣∣)∣∣∣ ~

w∣∣∣ w = [E : εa] · n−

[E :(

b+⊗

w)S]

n (5)

This can be rewritten as: f( ~

w)

∣∣∣ ~w∣∣∣ 1 + n·E·b+

·w = [E : εα] · n (6)

where 1 identifies the second-order identity tensor.By using this expression, the crack displacement can be solved for a certain set of

nodal displacements by means of the Newton–Raphson method. This model has beenimplemented by using a user subroutine UMAT of ABAQUS® and, since it needs informa-tion about the geometry of the model (the nodes coordinates of each element), it reads apreviously prepared external file.

2.2. Model Adopted by the MC2010

The MC2010 provides two possible constitutive laws to reproduce the behaviour ofFRC, with one describing a rigid-plastic behaviour and another one a linear behaviour.These models are defined by parameters that make reference to the values of the residualflexural tensile strength and the crack mouth opening displacement (CMOD) at specificinstants during a three-point bending test. In particular, the expression of the residualflexural tensile strength for a specific value of the CMOD may be obtained by expression (7):

fR,j =3Fjl

2bh2sp

(7)

where fR,j is the residual flexural tensile strength in MPa corresponding to a value ofthe CMOD = CMODj in mm, Fj is the load in N applied when CMOD = CMODj(in mm), l is the span length, b the specimen width and hsp the height of the notchedsection (ligament) where the crack propagates, with all these dimensions being in mm(see Figure 2b). Consequently, fR1 stands for the residual strength at a CMOD of 0.5 mmand fR3 stands for the residual strength at a CMOD of 2.5 mm. Figure 2a shows a typicalload-CMOD curve for FRC where some values of F and CMOD are identified accordingto the previous description, as described in the MC2010. Expression (7) can be easilyobtained by applying the basic principles of the strength of materials at the notched sectionof the specimen.

As mentioned before, MC2010 proposes two possible constitutive laws for FRC. Therigid-plastic model simply describes a constant stress behaviour from the point at whichcracking begins (w = 0) until it reaches a limit value (wu). In this model, the constant stressis obtained by considering the ultimate state when CMOD = CMOD3 and the ultimatevalue of w is considered as wu = CMOD3.

fFtu =fR3

3(8)

The linear model is defined by two reference values, fFts and fFtu, which can beobtained by (9) and (10):

fFts = 0.45 fR1 (9)

fFtu = fFts −wu

CMOD3( fFts − 0.5 fR3 + 0.2 fR1) ≥ 0 (10)

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(a)

(b)

Figure 2. (a) Typical load F-CMOD curve for FRC and (b) geometry of a three-point bending test specimen.

These values define a line which, as can be observed in these equations, is based on the experimental values for 𝐶𝑀𝑂𝐷 and 𝐶𝑀𝑂𝐷 . Neverheless, the ultimate crack opening wu is not specifically defined and is kept as the maximum crack opening accepted in struc-tural design, which depends on the ductility required.

Figure 3 shows the schemes for both models: the rigid-plastic and the linear model. It should be noted that while in this figure the linear model shows a load-descending be-haviour, depending on the performance of the PFRC, the linear model may present a pos-itive slope which in the MC2010 is referred to as post-crack hardening behaviour.

Figure 3. Rigid-plastic model (left) and linear model (right) as described by the MC2010.

2.3. Trilinear Model Proposed for PFRC In past papers published by the authors, the behaviour of hardened specimens of

self-compacting concrete (SCC) and vibrated conventional concrete (VCC) was studied, with particular examination of how the increase of fibre dosages affects the flexural be-haviour of specimens under three-point bending tests [35]. The authors proposed the use of a trilinear softening curve able to reproduce the classical three-point bending tests that develops under pure mode I conditions, as well as a modified version of the test proposed in [36], where cracking is initiated under a combination of modes (I) and (II) [37].

This trilinear model can be described by three linear parts, as can be seen in Figure 4. The first linear part reproduces the load decrease due to the failure of the concrete matrix up to the instant at which fibres make a relevant contribution to the specimen flexural strength. The second linear part reproduces the contribution of the fibres up to their

Figure 2. (a) Typical load F-CMOD curve for FRC and (b) geometry of a three-point bendingtest specimen.

These values define a line which, as can be observed in these equations, is based on theexperimental values for CMOD1 and CMOD3. Neverheless, the ultimate crack opening wuis not specifically defined and is kept as the maximum crack opening accepted in structuraldesign, which depends on the ductility required.

Figure 3 shows the schemes for both models: the rigid-plastic and the linear model.It should be noted that while in this figure the linear model shows a load-descendingbehaviour, depending on the performance of the PFRC, the linear model may present apositive slope which in the MC2010 is referred to as post-crack hardening behaviour.

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(a)

(b)

Figure 2. (a) Typical load F-CMOD curve for FRC and (b) geometry of a three-point bending test specimen.

These values define a line which, as can be observed in these equations, is based on the experimental values for 𝐶𝑀𝑂𝐷 and 𝐶𝑀𝑂𝐷 . Neverheless, the ultimate crack opening wu is not specifically defined and is kept as the maximum crack opening accepted in struc-tural design, which depends on the ductility required.

Figure 3 shows the schemes for both models: the rigid-plastic and the linear model. It should be noted that while in this figure the linear model shows a load-descending be-haviour, depending on the performance of the PFRC, the linear model may present a pos-itive slope which in the MC2010 is referred to as post-crack hardening behaviour.

Figure 3. Rigid-plastic model (left) and linear model (right) as described by the MC2010.

2.3. Trilinear Model Proposed for PFRC In past papers published by the authors, the behaviour of hardened specimens of

self-compacting concrete (SCC) and vibrated conventional concrete (VCC) was studied, with particular examination of how the increase of fibre dosages affects the flexural be-haviour of specimens under three-point bending tests [35]. The authors proposed the use of a trilinear softening curve able to reproduce the classical three-point bending tests that develops under pure mode I conditions, as well as a modified version of the test proposed in [36], where cracking is initiated under a combination of modes (I) and (II) [37].

This trilinear model can be described by three linear parts, as can be seen in Figure 4. The first linear part reproduces the load decrease due to the failure of the concrete matrix up to the instant at which fibres make a relevant contribution to the specimen flexural strength. The second linear part reproduces the contribution of the fibres up to their

Figure 3. Rigid-plastic model (left) and linear model (right) as described by the MC2010.

2.3. Trilinear Model Proposed for PFRC

In past papers published by the authors, the behaviour of hardened specimens of self-compacting concrete (SCC) and vibrated conventional concrete (VCC) was studied, withparticular examination of how the increase of fibre dosages affects the flexural behaviour ofspecimens under three-point bending tests [35]. The authors proposed the use of a trilinearsoftening curve able to reproduce the classical three-point bending tests that developsunder pure mode I conditions, as well as a modified version of the test proposed in [36],where cracking is initiated under a combination of modes (I) and (II) [37].

This trilinear model can be described by three linear parts, as can be seen in Figure 4.The first linear part reproduces the load decrease due to the failure of the concrete matrix upto the instant at which fibres make a relevant contribution to the specimen flexural strength.

Materials 2022, 15, 2323 6 of 15

The second linear part reproduces the contribution of the fibres up to their maximum tensilecapacity and in the third part the load decreases due to the final failure of fibres, which canbe due to fibre tensile failure or because of the slip of fibres inside the concrete.

Materials 2022, 15, x FOR PEER REVIEW 6 of 15

maximum tensile capacity and in the third part the load decreases due to the final failure of fibres, which can be due to fibre tensile failure or because of the slip of fibres inside the concrete.

Figure 4. Trilinear diagram proposed by the authors for PFRC.

2.4. Stress-Crack Opening Diagrams Used in This Study In order to compare the suitability of each of the three approaches described above,

this study examines the experimental behaviour of four PFRC mixes with the numerical predictions of each of them.

The reference experimental results correspond to four PFRC mixes of VCC with four fibre dosages: 3, 4.5, 6 and 10 kg/m3. These are selected results from a broader experi-mental campaign that studied several PFRC mixes and analysed their mechanical proper-ties through a series of two prismatic and nine cylindrical specimens of each mix; the de-tails regarding these experimental works may be consulted in [28,29].

Regarding the constitutive laws defined by the MC2010, which have been described before, Table 1 shows the parameters for each fibre dosage and for each of the two possi-bilities described in the standard: rigid-plastic model and linear model. Note that the lin-ear model is defined by two points: one for w = 0 (defined by) and another one defined by the value of w = CMOD . These diagrams are shown in Figure 5 and have been defined following the MC2010, thus a limit value of the crack opening of 2.5 mm has been em-ployed and Equations (8)–(10) used with the values of fR1 and fR3 experimentally obtained, which can be consulted in [28,29].

Table 1. Parameters of the rigid-plastic and the linear models for each fibre dosage of PFRC.

Rigid-Plastic Model Linear Model 𝒇𝑭𝒕𝒖 (MPa) 𝒘𝒖 (mm) 𝒇𝑭𝒕𝒔 (MPa) 𝒇𝑭𝒕𝒖 (MPa) 𝒘𝒖 (mm)

PFRC3 0.31 2.5 0.45 0.27 2.5 PFRC4.

5 0.49 2.5 0.55 0.48 2.5

PFRC6 0.79 2.5 0.70 0.88 2.5 PFRC10 0.98 2.5 0.88 1.06 2.5

As for the trilinear model, Table 2 shows the coordinates of the points that define them for each fibre dosage. The inverse analysis performed to obtain these values may be consulted in [35].

Figure 4. Trilinear diagram proposed by the authors for PFRC.

2.4. Stress-Crack Opening Diagrams Used in This Study

In order to compare the suitability of each of the three approaches described above,this study examines the experimental behaviour of four PFRC mixes with the numericalpredictions of each of them.

The reference experimental results correspond to four PFRC mixes of VCC withfour fibre dosages: 3, 4.5, 6 and 10 kg/m3. These are selected results from a broaderexperimental campaign that studied several PFRC mixes and analysed their mechanicalproperties through a series of two prismatic and nine cylindrical specimens of each mix;the details regarding these experimental works may be consulted in [28,29].

Regarding the constitutive laws defined by the MC2010, which have been describedbefore, Table 1 shows the parameters for each fibre dosage and for each of the two possibili-ties described in the standard: rigid-plastic model and linear model. Note that the linearmodel is defined by two points: one for w = 0 (defined by) and another one defined bythe value of w = CMOD3. These diagrams are shown in Figure 5 and have been definedfollowing the MC2010, thus a limit value of the crack opening of 2.5 mm has been employedand Equations (8)–(10) used with the values of fR1 and fR3 experimentally obtained, whichcan be consulted in [28,29].

Table 1. Parameters of the rigid-plastic and the linear models for each fibre dosage of PFRC.

Rigid-Plastic Model Linear Model

fFtu (MPa) wu (mm) fFts (MPa) fFtu (MPa) wu (mm)

PFRC3 0.31 2.5 0.45 0.27 2.5PFRC4.5 0.49 2.5 0.55 0.48 2.5PFRC6 0.79 2.5 0.70 0.88 2.5PFRC10 0.98 2.5 0.88 1.06 2.5

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Table 2. Coordinates of the points that define the trilinear model for each fibre dosage according to the point nomenclature provided in Figure 3.

T k r f σ (MPa) w (mm) σ (MPa) w (mm) σ (MPa) w (mm) σ(MPa) w (mm)

PFRC3 3.5 0 0.14 0.12 0.28 2.25 0 7.5 PFRC4.5 3.5 0 0.28 0.09 0.68 2.25 0 7.5 PFRC6 3.5 0 0.43 0.08 1.20 2.25 0 7.5 PFRC10 3.5 0 0.57 0.07 1.45 2.25 0 7.5

(a) (b)

(c) (d)

Figure 5. Comparison of the constitutive models obtained by each approach for different fibre dos-ages: (a) 3 kg/m³, (b) 4.5 kg/m³, (c) 6 kg/m³, (d) 10 kg/m³.

3. Numerical Simulations Numerical simulations were carried out by means of the finite element method with

the commercial software ABAQUS® (version 6.18). The softening behaviour was repro-duced by the embedded cohesive crack formulation described above, implemented through a user subroutine UMAT in ABAQUS®. Since, as required by the mathematical formulation described before, only linear triangular elements have been used, and the mesh is refined along the line where cracking develops (see Figure 6a); this refinement was designed based on previous works (see [37] and [38]) where the mesh dependence was already analysed.

In the case of models where fracture is reproduced by a trilinear model and by the linear model of the MC2010 with softening (negative slope of the diagram), all the ele-ments have been modelled by using the same cohesive formulation as that presented be-fore. In the case of those constitutive models with abrupt decrease of strength, like the rigid-plastic model and that termed “linear model with limited development”, conver-gence issues are usual when employing an implicit integration scheme. The convergence problem in the models that uses “linear softening with unlimited development” with hardening (positive slope of the diagram) provides a different explanation. In these cases,

Figure 5. Comparison of the constitutive models obtained by each approach for different fibredosages: (a) 3 kg/m3, (b) 4.5 kg/m3, (c) 6 kg/m3, (d) 10 kg/m3.

As for the trilinear model, Table 2 shows the coordinates of the points that definethem for each fibre dosage. The inverse analysis performed to obtain these values may beconsulted in [35].

Table 2. Coordinates of the points that define the trilinear model for each fibre dosage according tothe point nomenclature provided in Figure 3.

T k r f

σ (MPa) w (mm) σ (MPa) w (mm) σ (MPa) w (mm) σ (MPa) w (mm)

PFRC3 3.5 0 0.14 0.12 0.28 2.25 0 7.5PFRC4.5 3.5 0 0.28 0.09 0.68 2.25 0 7.5PFRC6 3.5 0 0.43 0.08 1.20 2.25 0 7.5

PFRC10 3.5 0 0.57 0.07 1.45 2.25 0 7.5

3. Numerical Simulations

Numerical simulations were carried out by means of the finite element method with thecommercial software ABAQUS® (version 6.18). The softening behaviour was reproducedby the embedded cohesive crack formulation described above, implemented through auser subroutine UMAT in ABAQUS®. Since, as required by the mathematical formulationdescribed before, only linear triangular elements have been used, and the mesh is refinedalong the line where cracking develops (see Figure 6a); this refinement was designed basedon previous works (see [37,38]) where the mesh dependence was already analysed.

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the material becomes harder with  increasing values of 𝑤,  (if an element  is damaged  it 

becomes harder, meaning that damage is more likely to develop around it instead of pro‐

gressing in the element that is already damaged). This leads to an increasing wider region 

of the specimen that develops, with it being more difficult for the model to find the crack 

path since, as load displacement increases, there are more possible elements to develop 

damage which means that convergence is not reached. Figure 6b shows one of these mod‐

els where damage started developing along the expected crack path but, at some point, 

started to develop around it and finally convergence was not reached when there were 

many possible elements to develop damage. 

(a)  (b) 

  

Figure 6. (a) Mesh refined along the crack path and boundary conditions, (b) maximum principal 

stress field in the damaged region of the model that used the linear constitutive law for 6 kg/m3 fibre 

dosage with no crack tracking. Convergence is not reached. 

In such cases with convergence problems, the path of fracture was tracked by defin‐

ing elements that could be damaged only in the crack path, with the rest of elements be‐

having elastically. 

Results 

Figure 7 shows the load‐deflection curves for each of the approaches described for 

various fibre dosages: 3, 4.5, 6 and 10 kg/m³, therefore reproducing the experimental re‐

sults of [28,29]. 

It is interesting to observe the differences among these approaches. In the case of the 

rigid‐plastic model, diagrams always show a plateau and a subsequent load decrease that 

describes an exponential‐like curve with small load jumps. As these drops are due to the 

abrupt load decrease described by the rectangular diagram, when an element reaches 𝑤𝑤 , there is a sudden load decrease. 

In  the case of  the “linear model with unlimited development”, the  load‐deflection 

diagrams exhibit a linear behaviour after the concrete matrix fails and, depending on the 

fibre dosage  the specimen, may show a softening behaviour  (for 3 and 4.5 kg/m3) or a 

hardening behaviour (for 6 and 10 kg/m3). Obviously, in the case of diagrams with hard‐

ening, this behaviour cannot be realistic, since the load may increase without a limit. Nev‐

ertheless,  it  is  interesting  to compare  these results with  those of  the  linear model with 

limited development that display an analogous behaviour until the most damaged ele‐

ment reaches 𝑤 𝑤 , when the load‐deflection diagrams show a highly similar behav‐

iour to that observed with the rigid‐plastic model. In all cases, the load drop follows an 

exponential‐like  shape with  small  jumps  that  can be  explained as  for  the  rigid‐plastic 

model. 

Lastly, if the results obtained with a trilinear model are observed, remarkable differ‐

ences can be highlighted compared with any of the results obtained with the other ap‐

proaches. First, the maximum load reached that identifies the beginning of the fracture 

process is higher than in any of the other cases and the shape shows an initial load decay, 

Figure 6. (a) Mesh refined along the crack path and boundary conditions, (b) maximum principalstress field in the damaged region of the model that used the linear constitutive law for 6 kg/m3 fibredosage with no crack tracking. Convergence is not reached.

In the case of models where fracture is reproduced by a trilinear model and by thelinear model of the MC2010 with softening (negative slope of the diagram), all the elementshave been modelled by using the same cohesive formulation as that presented before. Inthe case of those constitutive models with abrupt decrease of strength, like the rigid-plasticmodel and that termed “linear model with limited development”, convergence issues areusual when employing an implicit integration scheme. The convergence problem in themodels that uses “linear softening with unlimited development” with hardening (positiveslope of the diagram) provides a different explanation. In these cases, the material becomesharder with increasing values of w, (if an element is damaged it becomes harder, meaningthat damage is more likely to develop around it instead of progressing in the elementthat is already damaged). This leads to an increasing wider region of the specimen thatdevelops, with it being more difficult for the model to find the crack path since, as loaddisplacement increases, there are more possible elements to develop damage which meansthat convergence is not reached. Figure 6b shows one of these models where damage starteddeveloping along the expected crack path but, at some point, started to develop aroundit and finally convergence was not reached when there were many possible elements todevelop damage.

In such cases with convergence problems, the path of fracture was tracked by defin-ing elements that could be damaged only in the crack path, with the rest of elementsbehaving elastically.

Results

Figure 7 shows the load-deflection curves for each of the approaches described forvarious fibre dosages: 3, 4.5, 6 and 10 kg/m3, therefore reproducing the experimentalresults of [28,29].

It is interesting to observe the differences among these approaches. In the case ofthe rigid-plastic model, diagrams always show a plateau and a subsequent load decreasethat describes an exponential-like curve with small load jumps. As these drops are due tothe abrupt load decrease described by the rectangular diagram, when an element reachesw = wu, there is a sudden load decrease.

In the case of the “linear model with unlimited development”, the load-deflectiondiagrams exhibit a linear behaviour after the concrete matrix fails and, depending onthe fibre dosage the specimen, may show a softening behaviour (for 3 and 4.5 kg/m3) ora hardening behaviour (for 6 and 10 kg/m3). Obviously, in the case of diagrams withhardening, this behaviour cannot be realistic, since the load may increase without a limit.Nevertheless, it is interesting to compare these results with those of the linear model withlimited development that display an analogous behaviour until the most damaged elementreaches w = wu, when the load-deflection diagrams show a highly similar behaviour to that

Materials 2022, 15, 2323 9 of 15

observed with the rigid-plastic model. In all cases, the load drop follows an exponential-likeshape with small jumps that can be explained as for the rigid-plastic model.

Materials 2022, 15, x FOR PEER REVIEW 9 of 15

followed by a load recovery and another load decay later. Second, the last load decay, which identifies the last part of the test until failure, shows a curved shape that has the opposite curvature if compared with results obtained with the rigid-plastic model and the linear model with limited development.

(a) (b)

(c) (d)

Figure 7. Comparison of the load-deflection diagram obtained with each approach for different fibre dosages: (a) rigid-plastic model, (b) linear elastic model (unlimited development), (c) linear elastic model (limited development), (d) trilinear model.

4. Discussion In Figure 8 it can be seen that in the case of PFRC3 there are remarkable changes in

the shape of the curves obtained through using the post-peak constitutive functions ana-lysed. Regarding the rigid-plastic model and the linear model proposed by the MC2010, none is capable of reproducing the peak load registered in the experimental tests (it should be taken into account that usually 3 kg/m3 of polyolefin fibres do not allow PFRC to be classified as structural concrete). However, the trilinear approach accurately captures such a peak load. This phenomenon might be explained because the trilinear approach is the only one that establishes a progressive unloading after reaching peak load. Similarly, the trilinear approach is able to simulate the unloading process of the PFRC3 formulation. In the case of the rigid-plastic approach, when the deflection increases the load-deflection curve features a plateau. This stretch ends when a deflection of 3 mm is reached. From that deflection onwards, a progressive unloading takes place which does not fit the exper-imental behaviour. In the case of the “linear approach (unlimited)”, as the slope is negative

Figure 7. Comparison of the load-deflection diagram obtained with each approach for different fibredosages: (a) rigid-plastic model, (b) linear elastic model (unlimited development), (c) linear elasticmodel (limited development), (d) trilinear model.

Lastly, if the results obtained with a trilinear model are observed, remarkable dif-ferences can be highlighted compared with any of the results obtained with the otherapproaches. First, the maximum load reached that identifies the beginning of the fractureprocess is higher than in any of the other cases and the shape shows an initial load decay,followed by a load recovery and another load decay later. Second, the last load decay,which identifies the last part of the test until failure, shows a curved shape that has theopposite curvature if compared with results obtained with the rigid-plastic model and thelinear model with limited development.

4. Discussion

In Figure 8 it can be seen that in the case of PFRC3 there are remarkable changes in theshape of the curves obtained through using the post-peak constitutive functions analysed.Regarding the rigid-plastic model and the linear model proposed by the MC2010, none is

Materials 2022, 15, 2323 10 of 15

capable of reproducing the peak load registered in the experimental tests (it should be takeninto account that usually 3 kg/m3 of polyolefin fibres do not allow PFRC to be classifiedas structural concrete). However, the trilinear approach accurately captures such a peakload. This phenomenon might be explained because the trilinear approach is the only onethat establishes a progressive unloading after reaching peak load. Similarly, the trilinearapproach is able to simulate the unloading process of the PFRC3 formulation. In the case ofthe rigid-plastic approach, when the deflection increases the load-deflection curve featuresa plateau. This stretch ends when a deflection of 3 mm is reached. From that deflectiononwards, a progressive unloading takes place which does not fit the experimental behaviour.In the case of the “linear approach (unlimited)”, as the slope is negative the load decreasesas the deflection increases. Consequently, such simulation is able to maintain its load-bearing capacity beyond 6 mm of deflection. The results obtained with the “linear model(limited)” softening curve also boast a descending slope until a progressive unloading startsat 3 mm of deflection. From that point onwards, the curve is not capable of reproducingthe experimental behaviour. It should be underlined that the rigid-plastic approach andboth linear approaches obtain load–deflection curves that are above the experimental oneuntil a deflection of 2.5 mm is reached. For greater deflections, the experimental curve isabove that predicted by both linear approaches. In the case of the rigid-plastic model, asimilar comment could be made. However, in the case of the trilinear approach a closereproduction of the experimental behaviour PFRC3 is achieved throughout the test.

Materials 2022, 15, x FOR PEER REVIEW 10 of 15

the load decreases as the deflection increases. Consequently, such simulation is able to maintain its load-bearing capacity beyond 6 mm of deflection. The results obtained with the “linear model (limited)” softening curve also boast a descending slope until a progres-sive unloading starts at 3 mm of deflection. From that point onwards, the curve is not capable of reproducing the experimental behaviour. It should be underlined that the rigid-plastic approach and both linear approaches obtain load–deflection curves that are above the experimental one until a deflection of 2.5 mm is reached. For greater deflections, the experimental curve is above that predicted by both linear approaches. In the case of the rigid-plastic model, a similar comment could be made. However, in the case of the trilin-ear approach a close reproduction of the experimental behaviour PFRC3 is achieved throughout the test.

Figure 8. Comparison among the experimental curves and the simulations results obtained with the constitutive models implemented.

In the part of Figure 8 that corresponds to PFRC4.5, it may be observed that the dif-ferences perceived in the curves related with PFRC3 still appear. The peak load, as in the previous case, is not captured by any approach proposed by the MC2010 or even the mod-ification proposed for the linear model. However, the trilinear constitutive model is able to simulate the flexural tensile behaviour of the material prior to the development of the failure surface. As in the case of PFRC3, the trilinear softening function was able to simu-late the experimental behaviour of the PFRC4.5 in all its stages. Regarding the behaviour of the rigid-plastic model, it may be seen that its main feature is a constant load-bearing capacity until a 3 mm deflection is reached. From that point onwards the rigid-plastic model is below the experimental curve. As regards to both linear approaches are above

Figure 8. Comparison among the experimental curves and the simulations results obtained with theconstitutive models implemented.

Materials 2022, 15, 2323 11 of 15

In the part of Figure 8 that corresponds to PFRC4.5, it may be observed that thedifferences perceived in the curves related with PFRC3 still appear. The peak load, as inthe previous case, is not captured by any approach proposed by the MC2010 or even themodification proposed for the linear model. However, the trilinear constitutive model isable to simulate the flexural tensile behaviour of the material prior to the developmentof the failure surface. As in the case of PFRC3, the trilinear softening function was ableto simulate the experimental behaviour of the PFRC4.5 in all its stages. Regarding thebehaviour of the rigid-plastic model, it may be seen that its main feature is a constantload-bearing capacity until a 3 mm deflection is reached. From that point onwards therigid-plastic model is below the experimental curve. As regards to both linear approachesare above the experimental curve between 0.5 and 2.5 mm of deflection. There are fewdifferences between the rigid-plastic approach and the linear approach because the slope ofthe linear model is close to zero. One feature that could be highlighted is that the unlimitedlinear model behaviour is close to the experimental one for deflections between 2.5 and 6 mm.

Regarding the curves shown for PFRC6, a similar trend may be perceived if theexperimental curve is compared with the one obtained with the rigid-plastic approach. Fordeflections ranging from 0.3 mm to 2.5 mm, the simulated curve is above the experimentalone as the plateau is close to the experimental maximum post-peak load. From 2.5 mmof deflection, the unloading branch of the rigid-plastic model features a steep degree ofunloading that is below the experimental behaviour. Regarding the linear model with theunlimited option, it should be pointed out that as the slope of the load-deflection curve ispositive, the model is not able to capture the material failure as its load bearing capacityincreases with deflection. Thus, beyond 3.5 mm of deflection the linear model is abovethe experimental curve. This also occurs between 0.3 mm and 2.5 mm of deflection. If thelimited linear model is studied, similar comments to those previously mentioned regardingits unlimited version could be made until 2.5 mm of deflection is reached. From greaterdeflection values there are remarkable differences between the behaviour of the modeland the real one because steep unloading appears in the simulated curve. As regardsthe simulation performed with the trilinear approach, it could be said that it is able toreproduce the shape and values of the experimental curves although there are certaindifferences from 3.5 mm onwards. In this case, the experimental behaviour is slightly abovethe simulated one.

Regarding PFRC10, similar features to those observed in PFRC6 can be seen. Therigid-plastic approach has similar characteristics than those previously mentioned in thecase of PFRC6. A similar trend may be mentioned when the differences between theexperimental curve and both linear approaches are evaluated. On this occasion, the slope ofthe linear model is positive, and no failure of the material occurs. If the model is limited aunder evaluation of the behaviour of PFRC10 is observed. Regarding the trilinear approach,for deflection values greater than the maximum post-peak load such a curve marginallyunderestimates the load-bearing capacity of the material.

Figure 9 provides a comparison among the loads registered at different deflectionvalues in the simulations performed and takes into account the three constitutive modelsstudied. The values shown are relative to the value obtained in the experimental test whichis considered as unity. Consequently, if a load value obtained in the simulation is greaterthan the one obtained in the test, the column that appears in Figure 9 is greater than one.On the contrary, values lower than the unity reflect that the simulation underestimates thereal load-bearing capacity of the material. Such load analysis has been performed at threedeflections: 0.5 mm, 2.5 mm and 5 mm. The values of deflection studied are based on thealmost linear relation that exists between the crack width and the deflection. Consequently,it may be considered that a deflection value of 0.5 mm is close to the crack width used forchecking the SLS of the structural element. Correspondingly, the deflection value of 2.5 mmhas been chosen because such a value is close to the crack width chosen when analysingthe ultimate limit state of the structural element. Lastly, regarding the 5 mm of deflection,

Materials 2022, 15, 2323 12 of 15

such a value has been chosen as it may reflect the suitability of the constitutive models todetect the ultimate bearing capacity of the material at great deflections.

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Figure 9. Comparison of the loads borne by the simulations at certain deflection values with respect

to the experimental values.

When the analysis of the SLS is performed for the PFRC formulations, several

relevant features are evident. In the case of PFRC3, it may be seen that while the trilinear

approach underestimates the capacity of the material, the two proposals of the MC2010

boast loads greater than the experimental one. In the case of the linear model, both

versions obtain a load approximately 1.6 times the experimental one. In the case of the

rigid-plastic model, although it also overestimates the experimental value registered, it is

much closer to the real one. In the case of PFRC4.5, all the values obtained in the

simulations are greater than the experimental ones (except the trilinear one which is much

closer to the real value). On this occasion, the load values of the rigid-plastic model are

close to those obtained with the linear models. Similar results were obtained for the PFRC6

and PFRC10 formulations.

These features seem to be a consequence of the load-crack mouth opening

displacement curve taken as reference to set the requirements needed for considering the

contribution of fibres in FRC (see Figure 2a). As such a reference curve boasted an 𝑓𝑅3

value lower than fR1 if the design values are obtained by reducing the load obtained in

𝑓𝑅3, such a constitutive model might have an accurate correlation with material behaviour.

Nevertheless, in the case of PFRC, given that the value of 𝑓𝑅3 is greater than 𝑓𝑅1, the

design value obtained for reduced strains might not be an accurate reproduction of the

experimental behaviour.

When analysing the ULS, similar to the behaviour that appears at 2.5 mm of

deflection, it may be seen that the trilinear approach and those proposed by the MC2010

reflect a notable degree of accuracy with the experimental behaviour of the material.

Regarding the loads obtained with the trilinear approach, all load values are within

a 20% margin from the experimental at 5 mm of deflection. In the case of the rigid-plastic

model and the limited linear model, both are far from the experimental results for any

fibre dosage. In the case of the unlimited linear model, the load values are below the

experimental ones when the slope is negative. On the contrary, when the slope is positive

such values are slightly above those registered in the tests.

Another material property that might be of interest is the fracture energy required to

generate certain deflections of the concrete element. Consequently, the fracture energy at

the aforementioned deflections has been obtained for all the concrete formulations and

constitutive models analysed. Those results can be seen in Table 3, where the variation of

the fracture energy of the curves obtained in the simulations performed by using the

constitutive models implemented has been related to that obtained in the experimental

tests.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

PFRC3 PFRC4.5 PFRC6 PFRC10

Tri-linear

Rigid-plastic model

Linear model (un)

Linear model (lim)

Rela

tive l

oa

d v

alu

es

at

0.5

mm

defl

ect

ion

Concrete formulations

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

PFRC3 PFRC4.5 PFRC6 PFRC10

Tri-linear

Rigid-plastic model

Linear model (un)

Linear model (lim)

Rela

tive l

oa

d v

alu

es

at

2.5

mm

defl

ect

ion

Concrete formulations

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

PFRC3 PFRC4.5 PFRC6 PFRC10

Tri-linear

Rigid-plastic model

Linear model (un)

Linear model (lim)

Rela

tive l

oa

d v

alu

es

at

5m

m d

efl

ect

ion

Concrete formulations

Figure 9. Comparison of the loads borne by the simulations at certain deflection values with respectto the experimental values.

When the analysis of the SLS is performed for the PFRC formulations, several relevantfeatures are evident. In the case of PFRC3, it may be seen that while the trilinear approachunderestimates the capacity of the material, the two proposals of the MC2010 boast loadsgreater than the experimental one. In the case of the linear model, both versions obtain aload approximately 1.6 times the experimental one. In the case of the rigid-plastic model,although it also overestimates the experimental value registered, it is much closer to thereal one. In the case of PFRC4.5, all the values obtained in the simulations are greater thanthe experimental ones (except the trilinear one which is much closer to the real value). Onthis occasion, the load values of the rigid-plastic model are close to those obtained with thelinear models. Similar results were obtained for the PFRC6 and PFRC10 formulations.

These features seem to be a consequence of the load-crack mouth opening displace-ment curve taken as reference to set the requirements needed for considering the contribu-tion of fibres in FRC (see Figure 2a). As such a reference curve boasted an fR3 value lowerthan fR1 if the design values are obtained by reducing the load obtained in fR3, such a con-stitutive model might have an accurate correlation with material behaviour. Nevertheless,in the case of PFRC, given that the value of fR3 is greater than fR1, the design value obtainedfor reduced strains might not be an accurate reproduction of the experimental behaviour.

When analysing the ULS, similar to the behaviour that appears at 2.5 mm of deflection,it may be seen that the trilinear approach and those proposed by the MC2010 reflect anotable degree of accuracy with the experimental behaviour of the material.

Regarding the loads obtained with the trilinear approach, all load values are within a20% margin from the experimental at 5 mm of deflection. In the case of the rigid-plasticmodel and the limited linear model, both are far from the experimental results for anyfibre dosage. In the case of the unlimited linear model, the load values are below theexperimental ones when the slope is negative. On the contrary, when the slope is positivesuch values are slightly above those registered in the tests.

Another material property that might be of interest is the fracture energy required togenerate certain deflections of the concrete element. Consequently, the fracture energy atthe aforementioned deflections has been obtained for all the concrete formulations andconstitutive models analysed. Those results can be seen in Table 3, where the variation ofthe fracture energy of the curves obtained in the simulations performed by using the consti-tutive models implemented has been related to that obtained in the experimental tests.

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Table 3. Fracture energy consumption in the experimental tests and in the simulations.

Experimental Trilinear Rigid-Plastic Model Linear Model (un.) Linear Model (Lim.)

PFRC3 GF(N/mm)

GF(N/mm) ∆ (%) GF

(N/mm) ∆(%) GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%)

0.5 mm 0.160 0.147 −7.99 0.076 −52.55 0.106 −33.47 0.106 −33.472.5 mm 0.435 0.433 −0.60 0.387 −11.07 0.486 11.63 0.486 11.635 mm 0.849 0.846 −0.28 0.585 −31.07 0.829 −2.31 0.698 −17.75

PFRC4.5 GF(N/mm)

GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%) GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%)

0.5 mm 0.199 0.160 −19.68 0.118 −40.63 0.131 −34.23 0.131 −34.232.5 mm 0.609 0.582 −4.47 0.610 0.14 0.657 7.88 0.657 7.885 mm 1.239 1.210 −2.34 0.924 −25.43 1.262 1.83 0.978 −21.06

PFRC6 GF(N/mm)

GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%) GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%)

0.5 mm 0.196 0.176 −9.91 0.181 −7.42 0.164 −16.04 0.164 −16.042.5 mm 0.849 0.849 −0.04 0.979 15.34 0.941 10.88 0.941 10.885 mm 1.937 1.892 −2.33 1.488 −23.18 2.045 5.58 1.470 −24.11

PFRC10 GF(N/mm)

GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%) GF(N/mm) ∆ (%) GF

(N/mm) ∆ (%)

0.5 mm 0.184 0.193 5.22 0.221 20.24 0.202 10.23 0.202 10.232.5 mm 1.016 0.994 −2.10 1.206 18.69 1.155 13.76 1.155 13.765 mm 2.288 2.212 −3.33 1.844 −19.42 2.485 8.59 1.808 −21.01

In the case of the trilinear model, it can be seen that a notable precision was obtained atall the deflections analysed. Regarding the rigid-plastic model, there was a certain degreeof scattering between the values obtained at 0.5 mm between the experimental values andthe simulated ones. Such scattering was reduced at 2.5 mm of deflection and increased at5 mm of deflection. In the case of the linear model, the accuracy of the prediction of thefracture energy absorbed by the material increased as the deflection grew. In Table 3 it maybe seen that the proposed limitation of the linear approach did not enhance the accuracy ofthe prediction of the fracture energy absorbed.

5. Conclusions

The present contribution has implemented the constitutive relations proposed by theMC2010 for FRC in a user material subroutine seeking to assess the adaptability of theseconstitutive relations to PFRC. In addition, a trilinear model and an unlimited version ofthe linear model proposed by MC2010 have been employed as comparisons.

The simulations carried out with the implementation of the trilinear softening func-tion proposed by the authors showed a remarkable resemblance with the experimentalbehaviour of PFRC. Regarding the simulations performed through using the constitutivemodels proposed by MC2010, it was observed that they might not replicate the flexuralbehaviour of PFRC precisely. Neither the peak load nor the unloading process appeared inthe simulations performed with the rigid-plastic and linear constitutive relations.

When analysing the load values obtained at 0.5 mm of deflection, it is seen that theconstitutive relations proposed by the MC2010 are above the experimental results. AtULS, the relations proposed by the MC2010 and that proposed by the authors are able toreproduce with notable exactness the experimental load values. Regarding the ultimateload-bearing capacity of the material, which was assumed at a deflection of 5 mm, both thetrilinear and unlimited linear model were close to the load-bearing capacity of the material.On the contrary, the rigid-plastic model and the limited version of the linear model clearlyunderestimated the mechanical properties of PFRC.

Based on the aforementioned arguments, it seems that the rigid-plastic and the linearmodels proposed by the MC2010 might be employed with caution at SLS when applied

Materials 2022, 15, 2323 14 of 15

to PFRC. This might be a result of the limited degree of accuracy that the idealisationconsidered in MC2010 may offer when applied to FRC formulations where fR3 is notablygreater than fR1. The authors consider that if the results of the EN-14651 fracture testsboast a clear three-stretch behaviour after reaching the peak load (unloading-reloading-final unloading), a trilinear constitutive relation might be adopted as an alternative to themodels proposed by MC2010. This is relevant given that the latter models were developedconsidering that steel fibres were the main option for reinforcing concrete with fibres.

The limitation of this study might be that the differences detected were determined forcertain constitutive models stated in MC2010 and consequently if such models are modifiedthe present conclusions should be re-evaluated.

Author Contributions: Conceptualisation, A.E., F.S., M.G.A. and J.C.G.; methodology, A.E., F.S.,M.G.A. and J.C.G.; software, A.E., F.S. and J.C.G.; validation, A.E. and F.S.; formal analysis, M.G.A.and J.C.G.; investigation, A.E., F.S., M.G.A. and J.C.G.; resources, J.C.G.; data curation, A.E., F.S.,M.G.A. and J.C.G.; writing—original draft preparation, A.E. and F.S.; writing—review and editing,A.E., F.S., M.G.A. and J.C.G.; visualisation, A.E. and F.S.; supervision, J.C.G.; project administration,M.G.A.; funding acquisition, J.C.G. All authors have read and agreed to the published version ofthe manuscript.

Funding: This research was funded by the Ministry of Economy and Competitiveness of Spain bymeans of the Research Fund Project PID2019-108978RB-C31R.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The data presented in this study are available from the authorson request.

Acknowledgments: The authors wish to offer their gratitude to SIKA SAU for supporting theEnterprise University Chair “Cátedra Sika-UPM”.

Conflicts of Interest: The authors declare no conflict of interest.

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